Matrix factorization of a polynomial
In mathematics, a matrix factorization of a polynomial is a technique for factoring irreducible polynomials with matrices. David Eisenbud proved that every multivariate real-valued polynomial p without linear terms can be written as AB = pI, where A and B are square matrices and I is the identity matrix.[1] Given the polynomial p, the matrices A and B can be found by elementary methods.[2]
Example
[edit]The polynomial x2 + y2 is irreducible over R[x,y], but can be written as
![{\displaystyle \left[{\begin{array}{cc}x&-y\\y&x\end{array}}\right]\left[{\begin{array}{cc}x&y\\-y&x\end{array}}\right]=(x^{2}+y^{2})\left[{\begin{array}{cc}1&0\\0&1\end{array}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8c6602bba2981e7886239fc1eeebbc33006b361c)
References
[edit]- ^ Eisenbud, David (1980-01-01). "Homological algebra on a complete intersection, with an application to group representations". Transactions of the American Mathematical Society. 260 (1): 35. doi:10.1090/S0002-9947-1980-0570778-7. ISSN 0002-9947.
- ^ Crisler, David; Diveris, Kosmas, Matrix Factorizations of Sums of Squares Polynomials (PDF)
External links
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